Theorem fliftf 5794

Description: The domain and range of the function F. (Contributed by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
flift.1	|- F = ran ( x e. X |-> <. A , B >. )
flift.2	|- ( ( ph /\ x e. X ) -> A e. R )
flift.3	|- ( ( ph /\ x e. X ) -> B e. S )

Assertion

Ref	Expression
fliftf	|- ( ph -> ( Fun F <-> F : ran ( x e. X |-> A ) --> S ) )

Distinct variable groups:   x, R   ph, x   x, X   x, S

Allowed substitution hints:   A( x)   B( x)   F( x)

Proof of Theorem fliftf

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . . 5 |- ( ( ph /\ Fun F ) -> Fun F )
2		flift.1	. . . . . . . . . . 11 |- F = ran ( x e. X |-> <. A , B >. )
3		flift.2	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> A e. R )
4		flift.3	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> B e. S )
5	2, 3, 4	fliftel 5788	. . . . . . . . . 10 |- ( ph -> ( y F z <-> E. x e. X ( y = A /\ z = B ) ) )
6	5	exbidv 1825	. . . . . . . . 9 |- ( ph -> ( E. z y F z <-> E. z E. x e. X ( y = A /\ z = B ) ) )
7	6	adantr 276	. . . . . . . 8 |- ( ( ph /\ Fun F ) -> ( E. z y F z <-> E. z E. x e. X ( y = A /\ z = B ) ) )
8		rexcom4 2760	. . . . . . . . 9 |- ( E. x e. X E. z ( y = A /\ z = B ) <-> E. z E. x e. X ( y = A /\ z = B ) )
9		19.42v 1906	. . . . . . . . . . . 12 |- ( E. z ( y = A /\ z = B ) <-> ( y = A /\ E. z z = B ) )
10		elisset 2751	. . . . . . . . . . . . . 14 |- ( B e. S -> E. z z = B )
11	4, 10	syl 14	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. X ) -> E. z z = B )
12	11	biantrud 304	. . . . . . . . . . . 12 |- ( ( ph /\ x e. X ) -> ( y = A <-> ( y = A /\ E. z z = B ) ) )
13	9, 12	bitr4id 199	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> ( E. z ( y = A /\ z = B ) <-> y = A ) )
14	13	rexbidva 2474	. . . . . . . . . 10 |- ( ph -> ( E. x e. X E. z ( y = A /\ z = B ) <-> E. x e. X y = A ) )
15	14	adantr 276	. . . . . . . . 9 |- ( ( ph /\ Fun F ) -> ( E. x e. X E. z ( y = A /\ z = B ) <-> E. x e. X y = A ) )
16	8, 15	bitr3id 194	. . . . . . . 8 |- ( ( ph /\ Fun F ) -> ( E. z E. x e. X ( y = A /\ z = B ) <-> E. x e. X y = A ) )
17	7, 16	bitrd 188	. . . . . . 7 |- ( ( ph /\ Fun F ) -> ( E. z y F z <-> E. x e. X y = A ) )
18	17	abbidv 2295	. . . . . 6 |- ( ( ph /\ Fun F ) -> { y | E. z y F z } = { y | E. x e. X y = A } )
19		df-dm 4633	. . . . . 6 |- dom F = { y | E. z y F z }
20		eqid 2177	. . . . . . 7 |- ( x e. X |-> A ) = ( x e. X |-> A )
21	20	rnmpt 4871	. . . . . 6 |- ran ( x e. X |-> A ) = { y | E. x e. X y = A }
22	18, 19, 21	3eqtr4g 2235	. . . . 5 |- ( ( ph /\ Fun F ) -> dom F = ran ( x e. X |-> A ) )
23		df-fn 5215	. . . . 5 |- ( F Fn ran ( x e. X |-> A ) <-> ( Fun F /\ dom F = ran ( x e. X |-> A ) ) )
24	1, 22, 23	sylanbrc 417	. . . 4 |- ( ( ph /\ Fun F ) -> F Fn ran ( x e. X |-> A ) )
25	2, 3, 4	fliftrel 5787	. . . . . . 7 |- ( ph -> F C_ ( R X. S ) )
26	25	adantr 276	. . . . . 6 |- ( ( ph /\ Fun F ) -> F C_ ( R X. S ) )
27		rnss 4853	. . . . . 6 |- ( F C_ ( R X. S ) -> ran F C_ ran ( R X. S ) )
28	26, 27	syl 14	. . . . 5 |- ( ( ph /\ Fun F ) -> ran F C_ ran ( R X. S ) )
29		rnxpss 5056	. . . . 5 |- ran ( R X. S ) C_ S
30	28, 29	sstrdi 3167	. . . 4 |- ( ( ph /\ Fun F ) -> ran F C_ S )
31		df-f 5216	. . . 4 |- ( F : ran ( x e. X |-> A ) --> S <-> ( F Fn ran ( x e. X |-> A ) /\ ran F C_ S ) )
32	24, 30, 31	sylanbrc 417	. . 3 |- ( ( ph /\ Fun F ) -> F : ran ( x e. X |-> A ) --> S )
33	32	ex 115	. 2 |- ( ph -> ( Fun F -> F : ran ( x e. X |-> A ) --> S ) )
34		ffun 5364	. 2 |- ( F : ran ( x e. X |-> A ) --> S -> Fun F )
35	33, 34	impbid1 142	1 |- ( ph -> ( Fun F <-> F : ran ( x e. X |-> A ) --> S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   E.wex 1492   e. wcel 2148   {cab 2163   E.wrex 2456   C_ wss 3129   <.cop 3594   class class class wbr 4000   |-> cmpt 4061   X. cxp 4621   dom cdm 4623   ran crn 4624   Fun wfun 5206   Fn wfn 5207   -->wf 5208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220

This theorem is referenced by:  qliftf  6614